AsymptoticGreater

AsymptoticGreater[f,g,xx*]

gives conditions for or as xx*.

AsymptoticGreater[f,g,{x1,,xn}{,,}]

gives conditions for or as .

Details and Options

  • Asymptotic greater is also expressed as f is little-omega of g, f is of order greater than g, f grows faster than g, and f dominates g. The point x* is often assumed from context.
  • Asymptotic greater is an order relation and means TemplateBox[{{f, (, x, )}}, Abs]>=c TemplateBox[{{g, (, x, )}}, Abs] when x is near x* for all constants .
  • Typical uses include expressing simple strict lower bounds for functions and sequences near some point. It is frequently used for solutions to equations and to give simple strict lower bounds for computational complexity.
  • For a finite limit point x* and {,,}:
  • AsymptoticGreater[f[x],g[x],xx*]for all there exists such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(c,x^*) implies TemplateBox[{{f, (, x, )}}, Abs]>=c TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticGreater[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]for all there exists such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]>=c TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • For an infinite limit point:
  • AsymptoticGreater[f[x],g[x],x]for all there exists such that implies TemplateBox[{{f, (, x, )}}, Abs]>=c TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticGreater[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]for all there exists such that implies TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]>=c TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • AsymptoticGreater[f[x],g[x],xx*] exists if and only if Limit[Abs[f[x]/g[x]],xx*] when g[x] does not have an infinite set of zeros near x*.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DirectionRealsdirection to approach the limit point
    GenerateConditionsAutomaticgenerate conditions for parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"what to optimize
  • Possible settings for Direction include:
  • Reals or "TwoSided"from both real directions
    "FromAbove" or -1from above or larger values
    "FromBelow" or +1from below or smaller values
    Complexesfrom all complex directions
    Exp[ θ]in the direction
    {dir1,,dirn}use direction diri for variable xi independently
  • DirectionExp[ θ] at x* indicates the direction tangent of a curve approaching the limit point x*.
  • Possible settings for GenerateConditions include:
  • Automaticnongeneric conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticGreater typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (2)

Verify that as :

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Visualize the two functions:

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Verify that as :

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Visualize the two functions:

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Scope  (9)

Options  (10)

Applications  (18)

Properties & Relations  (10)

See Also

AsymptoticGreaterEqual  AsymptoticLess  AsymptoticLessEqual  AsymptoticEqual  AsymptoticEquivalent  Limit

Introduced in 2018
(11.3)